Real-rooted atlas

A091042 → basis A013609 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-6 + 24\,x - 6\,x^{2} + 19\,n - 28\,n x + 4\,n x^{2} - 16\,n^{2} + 8\,n^{2} x + 4\,n^{3}\right)\,P_{n}(x) = \left(24 + 108\,x + 84\,x^{2} - 36\,x^{3} + 20\,n - 54\,n x - 128\,n x^{2} + 24\,n x^{3} - 48\,n^{2} - 48\,n^{2} x + 48\,n^{2} x^{2} + 16\,n^{3} + 24\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x + 28\,x^{2} + 24\,x^{3} - 16\,x^{4} + 8\,n x - 32\,n x^{3} - 8\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
210616
3568020156
4288672392561408
5140846084608153614412304
666562816040832239365280352105216
73072015974430617626624010483216640832885184
81392648601602058240240640014284804085764928019207352320

All 15 rows

Technical details
Node id
n-d634ae1414b25970 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-925e1f55d72de84d

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:      10,        6;
   3:      56,       80,       20;
   4:     288,      672,      392,        56;
   5:    1408,     4608,     4608,      1536,       144;
   6:    6656,    28160,    40832,     23936,      5280,      352;
   7:   30720,   159744,   306176,    266240,    104832,    16640,      832;
   8:  139264,   860160,  2058240,   2406400,   1428480,   408576,    49280,    1920;
   9:  622592,  4456448, 12812288,  18939904,  15319040,  6684672,  1462272,  139264,   4352;
  10: 2752512, 22413312, 75333632, 135102464, 139616256, 83894272, 28250112, 4902912, 379392, 9728;
%F Formula
The row polynomials satisfy (4*n^3-16*n^2+19*n-6+(8*n^2-28*n+24)*x+(4*n-6)*x^2)*P(n,x) = (16*n^3-48*n^2+20*n+24+(24*n^3-48*n^2-54*n+108)*x+(48*n^2-128*n+84)*x^2+(24*n-36)*x^3)*P(n-1,x) - ((8*n^2-8*n-6)*x+(16*n^2-28)*x^2+(32*n-24)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 48*n^2 + 20*n + 24 + (24*n^3 - 48*n^2 - 54*n + 108)*x + (48*n^2 - 128*n + 84)*x^2 + (24*n - 36)*x^3)*P[n - 1, x] - ((8*n^2 - 8*n - 6)*x + (16*n^2 - 28)*x^2 + (32*n - 24)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(4*n^3 - 16*n^2 + 19*n - 6 + (8*n^2 - 28*n + 24)*x + (4*n - 6)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n^3 - 48*n^2 + 20*n + 24 + (24*n^3 - 48*n^2 - 54*n + 108)*x + (48*n^2 - 128*n + 84)*x^2 + (24*n - 36)*x^3)*v[i - 1] - ((8*n^2 - 8*n - 6)*x + (16*n^2 - 28)*x^2 + (32*n - 24)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(4*n^3 - 16*n^2 + 19*n - 6 + (8*n^2 - 28*n + 24)*x + (4*n - 6)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023